Describing Data

Summary Statistics, Distributions, and Why Pictures Matter

Learning Objectives

  1. Calculate and interpret mean, median, mode
  2. Recognise when the mean misleads and the median is better
  3. Quantify variability: range, IQR, SD, CV
  4. Describe the Normal distribution and the 68-95-99.7 rule
  5. Identify skewed distributions and choose appropriate summaries
  6. Read and interpret box plots, violin plots, histograms
  7. Explain why summary stats alone can deceive (Anscombe’s quartet)

Clinical Hook: The “Average” Doctor’s Salary

Group A (Government college): Everyone earns ~₹1,00,000/month

Group B (Private): 8 juniors at ₹80,000, 1 senior at ₹3,00,000, 1 star surgeon at ₹12,00,000

  • Group A mean: ₹1,00,000
  • Group B mean: ₹2,10,000
  • Group B median: ₹80,000

The mean of ₹2,10,000 represents nobody in Group B.

Now replace “salary” with serum creatinine in a renal OPD. Same story — a few CKD-5 patients pull the mean far from reality.

Measures of Central Tendency

Measure Formula Best for Affected by outliers?
Mean Sum / n Symmetric data Yes — heavily
Median Middle value Skewed data No — robust
Mode Most frequent Categorical data No

Quick Rule

If mean ≠ median, the data is skewed. The direction of the skew is toward the mean.

  • Mean > Median → Right-skewed (e.g., income, CRP, LOS)
  • Mean < Median → Left-skewed (rare in clinical data)

Worked Example: HbA1c

Figure 1

Measures of Dispersion

For symmetric data:

  • Range = Max - Min (sensitive to outliers)
  • Standard Deviation (SD) = average distance from mean
  • Variance = SD²

\[SD = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}\]

For skewed data:

  • IQR = Q3 - Q1 (middle 50%)
  • Percentiles (e.g., 5th, 95th)

Comparing scales:

  • CV = (SD / Mean) × 100%
  • Useful when comparing variability of height (cm) vs weight (kg)

The Normal Distribution: 68-95-99.7 Rule

Figure 2

Clinical Application: Serum Sodium

Population: Healthy adults, Mean = 140 mEq/L, SD = 3 mEq/L

Rule Range %
Mean ± 1 SD 137–143 68%
Mean ± 2 SD 134–146 95%
Mean ± 3 SD 131–149 99.7%

Reference Range = Mean ± 2 SD

The “normal range” on your lab report (134–146 mEq/L) is literally mean ± 2 SD. Values outside this range are in the extreme 5% — flagged as abnormal.

When Data Is NOT Normal: Skewness

Figure 3

Right-skewed clinical examples: Hospital LOS, CRP, income, bilirubin, serum creatinine (mixed populations)

Box Plot vs Violin Plot: CRP by Sepsis Status

Figure 4

Anscombe’s Quartet: Never Trust Numbers Alone

Figure 5

Choosing Summary Statistics

Distribution Centre Spread Report as
Symmetric / Normal Mean SD Mean ± SD
Skewed Median IQR Median (IQR)
Ordinal Median IQR Median (IQR)
Categorical Mode Frequency n (%)

The #1 Mistake in Medical Papers

Reporting mean ± SD for skewed data (hospital LOS, CRP, costs). Always check the distribution shape first!

Key Takeaways

  1. Mean vs Median: skewed data → use the median (creatinine, CRP, LOS)

  2. SD vs IQR: SD for symmetric; IQR for skewed distributions

  3. 68-95-99.7 rule: the foundation of clinical reference ranges

  4. Always visualise: Anscombe’s quartet proves numbers alone can deceive

  5. Box + violin = best of both worlds for group comparisons

Further Learning

Videos:

Books:

  • Bland M. An Introduction to Medical Statistics. Ch 4–5.
  • Kirkwood BR. Essential Medical Statistics. Ch 2–4.